Worked example 1
Solve with an initial condition
Given dy/dx = 6x − 4 and y = 5 when x = 1, find y in terms of x
- Integrate to get y = 3x² − 4x + C
- Substitute x = 1 and y = 5
- Solve −1 + C = 5.
Answer: y = 3x² − 4x + 6
Higher Mathematics
Higher Maths Differential equations and rates
Practise differential equations and rates for Scottish Higher Mathematics with worked examples, clear methods and original interactive questions. Solve simple equations of the form dy/dx = f(x) and interpret initial-condition models.
Before you start
- Integrate algebraic and trig functions.
- Use an initial condition.
- Interpret variables and units.
Explanation
At Higher, a simple differential equation gives dy/dx directly as a function of x. Integrate to obtain the family of possible functions.
An initial condition fixes the constant, and the resulting model can then answer contextual questions.
Method and rules
- If dy/dx = f(x), then y = integral f(x) dx + C.
- Use y(x₀) = y₀ to determine C.
- Retain units and interpret the final value in context.
Worked examples
Worked example 2
Method check
Why is an initial condition needed?
- Many functions share the same derivative.
So: It determines the constant of integration and selects one function.
Watch out
- Stopping with an unknown C after an initial condition is given.
- Substituting the condition into dy/dx rather than y
- Dropping contextual units from a rate.
Exam reminder
For differential equations and rates, show the defining equation or formula before simplifying. Check that every final value satisfies the original restrictions, interval or context.
Related Higher Maths topics
Continue with the methods that connect most closely to this topic.